Is it possible to obtain the general solution of an ODE by solving a PDE, SDE or SPDE? I haven’t dived into PDEs, nor SDEs, therefore the question.

Another question that arises from my love to differential equations and love for general cases. From a higher mathematical perspective, how PDEs and SPDEs can be generalized? I was thinking of a more general operator that behaves like the differential operator in certrain structures, instead of use real numbers use hypercomplex numbers (from the Cayley–Dickson construction or Clifford algebras) or things a lot more general that I don’t know. Instead of limiting to apply the operator once, twice or thrice extend it in a way that it is possible to apply a hypercomplex number of times (if the Gamma function were involved, for example, there’s a way to extend it to hypercomplex numbers from the Cayley–Dickson construction, via defining the Gamma function of a matrix). Hence, what is the most general kind of differential equation that you know? and, what is the most general kind of (not necessarily differential) operator that you know?

hypercomplex numberslie groupsordinary differential equationspartial differential equationsstochastic differential equations

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